The description of snow microstructure in microwave models is often simplified to facilitate electromagnetic calculations. Within dense media radiative transfer (DMRT), the microstructure is commonly described by sticky hard spheres (SHS). An objective mapping of real snow onto SHS is however missing which prevents measured input parameters from being used for DMRT. In contrast, the microwave emission model of layered snowpacks (MEMLS) employs a conceptually different approach, based on the two-point correlation function which is accessible by tomography. Here we show the equivalence of both electromagnetic approaches by reformulating their microstructural models in a common framework. Using analytical results for the two-point correlation function of hard spheres, we show that the scattering coefficient in both models only differs by a factor which is close to unity, weakly dependent on ice volume fraction and independent of other microstructural details. Additionally, our analysis provides an objective retrieval method for the SHS parameters (diameter and stickiness) from tomography images. For a comprehensive data set we demonstrate the variability of stickiness and compare the SHS diameter to the optical equivalent diameter. Our results confirm the necessity of a large grain-size scaling when relating both diameters in the non-sticky case, as previously suggested by several authors.

Microwave modeling of snow is commonly addressed within multilayer
approaches to account for the vertical layer structure of a
snowpack. Examples are HUT

Electromagnetic properties of single snow layers can be theoretically
obtained by homogenization methods, if snow is treated as a random,
two-phase medium which is statistically homogeneous. The microwave emission model
of layered snowpacks (MEMLS) is based on the improved Born approximation (IBA)

Advantages and disadvantages of IBA have to be discussed at eye level
of those inherent to another electromagnetic approach to scattering in
snow, namely dense media radiative transfer (DMRT). Originally DMRT
was developed for random media consisting of spheres or
spheroids. Sphere models are attractive from various
perspectives. First, the scattering coefficient of sphere assemblies
can be calculated analytically in various approximation schemes, such
as the quasi-crystalline approximation (QCA), which can be optionally
improved by the so-called coherent potential (QCA-CP)

Objective means of estimating the stickiness parameter for a given snow
sample are hitherto missing. However, simply resorting to the
non-sticky case causes other difficulties. Recent DMRT-based microwave
emission modeling

It is the aim of the present paper to advance the understanding of
microstructural models in different electromagnetic models for
microwave modeling of snow by establishing a rigorous link between the
scattering formulations used in DMRT-ML and MEMLS. More precisely, for
the DMRT theory we consider the QCA-CP approximation as used in
DMRT-ML

The paper is organized as follows. In Sect.

Random two-phase media are a natural starting point to characterize
the morphology of air and ice in snow. We consider a two-phase medium
in a region

Following

Though snow is known to be anisotropic

Below we state the governing equations for the scattering coefficient
in the improved Born approximation

Within the improved Born approximation, the scattering coefficient is
derived from the phase function (or bistatic scattering function)

We close this section by commenting on the ambiguous notion of
“particles” in the original derivation of IBA. The field ratio
matrix

The scattering coefficient

The original versions of DMRT assume that the microstructure comprises

Several flavors of DMRT have been developed over the years

Different strategies are possible to solve Eq. (

The statistical characterization of particle systems by the number
density field Eq. (

The interrelation of the statistical description of two-phase media
and particle-based media in terms of correlation functions was
detailed by

The link between corresponding correlation functions was established

By means of the fundamental relation Eq. (

Ratio of scattering coefficient calculated with IBA and DMRT theory
as a function of ice volume fraction

The DMRT formulation used in many DMRT-based models use the sticky
hard sphere (SHS) model to represent the position of the
scatterers. In IBA it is now possible to consider the same SHS model
since the scattering coefficient Eq. (

Real and imaginary parts of dielectric constant

In summary, the Fourier transform of the correlation function of
sticky hard spheres can be written as

For the following analysis we employ the data set of

Computing correlation functions as convolutions of

Snow is known to be anisotropic which was explicitly analyzed for the
present data set in

Comparison of measured and fitted Fourier transforms of the
correlation function for one snow sample (blue curves). Dependence of sticky
hard sphere model

Before turning to the parameter estimation, we illustrate the
parametric behavior of

To further demonstrate the impact of the parameters (

Using the closed form expression

Estimated parameter pairs (

For an overview, we show the optimal parameters (

Next we plot the optimal stickiness values

Estimated stickiness values

The second (full) line in Fig.

Finally we compare the estimated SHS diameter

To illustrate differences in the performance of the fit for the SHS
model, we show the coefficient of determination

Scatter plot of the SHS diameter estimate

To further investigate goodness-of-fit differences of the SHS model, we
analyzed the behavior of the cost function

In order to assess the relevance of the grain-size scaling raised
in

Fit parameters and standard errors for a linear regression between the optimal
SHS diameter and the optical diameter according to Eq. (

Coefficient of determination

As a quantitative measure, we fitted the entire data in
Fig.

Contour plots of the root-mean-square error surface for different snow types. Colors
show the logarithm of the sum of the squared differences between measured and
parametric SHS form

Comparison of the

In addition, we conducted a numerical experiment to reproduce the
situation from

With the set of optimal parameters (

The results are shown in Fig.

Scatter plot of the scattering coefficient of IBA and QCA-CP from
Eqs. (

Three main implications can be drawn from the present work.

The first is related to the comparison of the electromagnetic models
IBA used in MEMLS and QCA-CP used in DMRT-ML. By rederiving the
scattering coefficient in IBA and QCA-CP and extracting the dependence
on microstructure we have shown that both electromagnetic
approximations involve exactly the same microstructural
characteristic, namely the zero-wavevector component of the structure
factor

The second implication of the present work is related to parameter
estimation itself. The closed form expression of the correlation
function (or its FFT) for the monodisperse SHS model allows
parameters (

Fit parameters and standard errors for a linear regression
for all samples between the optimal SHS diameter and the optical diameter if the
optimization of

The third implication of the work is related to the applicability of
what has been termed the “short range limit” in microwave models

One main motivation for the present work was the issue of grain-size
scaling raised by

The SSA, or optical radius, of snow can be easily measured in the
field and is an appealing first guess as the size parameter in DMRT-based
sphere models. However, all simulations using

Figure

When stickiness is not known in advance, but set to a fixed,
prescribed value, our numerical experiment (Table

A peculiar feature of the monodisperse SHS correlation function

The monodisperse version of SHS bears another peculiarity. In general,
the Fourier transform of the correlation function must reveal the SSA
in its large

If, however, the present method of

To assess the goodness-of-fit of monodisperse SHS, we have evaluated
the coefficient of determination

The performance of any correlation function model has to be assessed
against microwave measurements which eventually decide about the
quality of a particular model. We have shown that in both scattering
formulations, IBA and QCA-CP, it comes down to a single
microstructural quantity which must be well-predicted to describe
scattering correctly in the low frequency limit. This quantity is the
integral of the correlation function, or likewise, the zero-wavevector
component

In view of the future task of finding the best microstructural model
for microwave modeling, we suggest building on the exhaustive work on
small angle scattering used for molecular systems. Our reanalysis has
stressed that the relevant quantity in IBA and QCA-CP in the
scattering coefficient for microwave modeling of snow is the Fourier
transform of the correlation function which must be well-matched. This
task is well-known and completely analogous to small angle scattering (SAS)
of molecular systems. SAS from X-ray or neutron sources has
become a standard technique to characterize microstructures by fitting
Fourier data

The results about snow as a particle-based (granular) medium gained
from the present work can be exploited even beyond the context of
microwaves. As an example, discrete element modeling (DEM) is of
special interest for snow mechanics

We reformulated two relevant approaches to the microwave scattering
coefficient of snow, namely IBA used in MEMLS and QCA-CP used in
DMRT-ML, in a common microstructural framework. This revealed their
quasi-equivalence when using the same microstructural, particle-based
model. As an implication of the theoretical analysis, the stickiness
parameter for (monodisperse) SHS can now be objectively estimated from

We thank M. Proksch for helpful comments on the manuscript. G. Picard was sponsored by a grant from the Institut Universitaire de France and by the CNES-TOSCA SMOS project. Edited by: R. Brown